Calculus Examples

$\frac{d z}{d x} + 2 z = 4 x$

Step 1

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = 2$ .

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Step 1.1

Set up the integration.

$e^{\int 2 d x}$

Step 1.2

Apply the constant rule.

$e^{2 x + C}$

Step 1.3

Remove the constant of integration.

$e^{2 x}$

Step 2

Multiply each term by the integrating factor $e^{2 x}$ .

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Step 2.1

Multiply each term by $e^{2 x}$ .

$e^{2 x} \frac{d z}{d x} + e^{2 x} (2 z) = e^{2 x} (4 x)$

Step 2.2

Rewrite using the commutative property of multiplication.

$e^{2 x} \frac{d z}{d x} + 2 e^{2 x} z = e^{2 x} (4 x)$

Step 2.3

Rewrite using the commutative property of multiplication.

$e^{2 x} \frac{d z}{d x} + 2 e^{2 x} z = 4 e^{2 x} x$

Step 2.4

Reorder factors in $e^{2 x} \frac{d z}{d x} + 2 e^{2 x} z = 4 e^{2 x} x$ .

$e^{2 x} \frac{d z}{d x} + 2 z e^{2 x} = 4 x e^{2 x}$

Step 3

Rewrite the left side as a result of differentiating a product.

$\frac{d}{d x} [e^{2 x} z] = 4 x e^{2 x}$

Step 4

Set up an integral on each side.

$\int \frac{d}{d x} [e^{2 x} z] d x = \int 4 x e^{2 x} d x$

Step 5

Integrate the left side.

$e^{2 x} z = \int 4 x e^{2 x} d x$

Step 6

Integrate the right side.

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Step 6.1

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$e^{2 x} z = 4 \int x e^{2 x} d x$

Step 6.2

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = e^{2 x}$ .

$e^{2 x} z = 4 (x (\frac{1}{2} e^{2 x}) - \int \frac{1}{2} e^{2 x} d x)$

Step 6.3

Simplify.

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Step 6.3.1

Combine $\frac{1}{2}$ and $e^{2 x}$ .

$e^{2 x} z = 4 (x \frac{e^{2 x}}{2} - \int \frac{1}{2} e^{2 x} d x)$

Step 6.3.2

Combine $x$ and $\frac{e^{2 x}}{2}$ .

$e^{2 x} z = 4 (\frac{x e^{2 x}}{2} - \int \frac{1}{2} e^{2 x} d x)$

Step 6.3.3

Combine $\frac{1}{2}$ and $e^{2 x}$ .

$e^{2 x} z = 4 (\frac{x e^{2 x}}{2} - \int \frac{e^{2 x}}{2} d x)$

Step 6.4

Since $\frac{1}{2}$ is constant with respect to $x$ , move $\frac{1}{2}$ out of the integral.

$e^{2 x} z = 4 (\frac{x e^{2 x}}{2} - (\frac{1}{2} \int e^{2 x} d x))$

Step 6.5

Let $u = 2 x$ . Then $d u = 2 d x$ , so $\frac{1}{2} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 6.5.1

Let $u = 2 x$ . Find $\frac{d u}{d x}$ .

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Step 6.5.1.1

Differentiate $2 x$ .

$\frac{d}{d x} [2 x]$

Step 6.5.1.2

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$2 \frac{d}{d x} [x]$

Step 6.5.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$2 \cdot 1$

Step 6.5.1.4

Multiply $2$ by $1$ .

$2$

Step 6.5.2

Rewrite the problem using $u$ and $d u$ .

$e^{2 x} z = 4 (\frac{x e^{2 x}}{2} - \frac{1}{2} \int e^{u} \frac{1}{2} d u)$

Step 6.6

Combine $e^{u}$ and $\frac{1}{2}$ .

$e^{2 x} z = 4 (\frac{x e^{2 x}}{2} - \frac{1}{2} \int \frac{e^{u}}{2} d u)$

Step 6.7

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$e^{2 x} z = 4 (\frac{x e^{2 x}}{2} - \frac{1}{2} (\frac{1}{2} \int e^{u} d u))$

Step 6.8

Simplify.

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Step 6.8.1

Multiply $\frac{1}{2}$ by $\frac{1}{2}$ .

$e^{2 x} z = 4 (\frac{x e^{2 x}}{2} - \frac{1}{2 \cdot 2} \int e^{u} d u)$

Step 6.8.2

Multiply $2$ by $2$ .

$e^{2 x} z = 4 (\frac{x e^{2 x}}{2} - \frac{1}{4} \int e^{u} d u)$

Step 6.9

The integral of $e^{u}$ with respect to $u$ is $e^{u}$ .

$e^{2 x} z = 4 (\frac{x e^{2 x}}{2} - \frac{1}{4} (e^{u} + C))$

Step 6.10

Rewrite $4 (\frac{x e^{2 x}}{2} - \frac{1}{4} (e^{u} + C))$ as $4 (\frac{1}{2} x e^{2 x} - \frac{1}{4} e^{u}) + C$ .

$e^{2 x} z = 4 (\frac{1}{2} x e^{2 x} - \frac{1}{4} e^{u}) + C$

Step 6.11

Replace all occurrences of $u$ with $2 x$ .

$e^{2 x} z = 4 (\frac{1}{2} x e^{2 x} - \frac{1}{4} e^{2 x}) + C$

Step 6.12

Simplify.

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Step 6.12.1

Simplify each term.

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Step 6.12.1.1

Combine $\frac{1}{2}$ and $x$ .

$e^{2 x} z = 4 (\frac{x}{2} e^{2 x} - \frac{1}{4} e^{2 x}) + C$

Step 6.12.1.2

Combine $\frac{x}{2}$ and $e^{2 x}$ .

$e^{2 x} z = 4 (\frac{x e^{2 x}}{2} - \frac{1}{4} e^{2 x}) + C$

Step 6.12.1.3

Combine $e^{2 x}$ and $\frac{1}{4}$ .

$e^{2 x} z = 4 (\frac{x e^{2 x}}{2} - \frac{e^{2 x}}{4}) + C$

Step 6.12.2

Apply the distributive property.

$e^{2 x} z = 4 \frac{x e^{2 x}}{2} + 4 (- \frac{e^{2 x}}{4}) + C$

Step 6.12.3

Cancel the common factor of $2$ .

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Step 6.12.3.1

Factor $2$ out of $4$ .

$e^{2 x} z = 2 (2) \frac{x e^{2 x}}{2} + 4 (- \frac{e^{2 x}}{4}) + C$

Step 6.12.3.2

Cancel the common factor.

$e^{2 x} z = 2 \cdot 2 \frac{x e^{2 x}}{2} + 4 (- \frac{e^{2 x}}{4}) + C$

Step 6.12.3.3

Rewrite the expression.

$e^{2 x} z = 2 (x e^{2 x}) + 4 (- \frac{e^{2 x}}{4}) + C$

Step 6.12.4

Cancel the common factor of $4$ .

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Step 6.12.4.1

Move the leading negative in $- \frac{e^{2 x}}{4}$ into the numerator.

$e^{2 x} z = 2 (x e^{2 x}) + 4 \frac{- e^{2 x}}{4} + C$

Step 6.12.4.2

Cancel the common factor.

$e^{2 x} z = 2 (x e^{2 x}) + 4 \frac{- e^{2 x}}{4} + C$

Step 6.12.4.3

Rewrite the expression.

$e^{2 x} z = 2 (x e^{2 x}) - e^{2 x} + C$

$e^{2 x} z = 2 x e^{2 x} - e^{2 x} + C$

Step 7

Divide each term in $e^{2 x} z = 2 x e^{2 x} - e^{2 x} + C$ by $e^{2 x}$ and simplify.

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Step 7.1

Divide each term in $e^{2 x} z = 2 x e^{2 x} - e^{2 x} + C$ by $e^{2 x}$ .

$\frac{e^{2 x} z}{e^{2 x}} = \frac{2 x e^{2 x}}{e^{2 x}} + \frac{- e^{2 x}}{e^{2 x}} + \frac{C}{e^{2 x}}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $e^{2 x}$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{e^{2 x} z}{e^{2 x}} = \frac{2 x e^{2 x}}{e^{2 x}} + \frac{- e^{2 x}}{e^{2 x}} + \frac{C}{e^{2 x}}$

Step 7.2.1.2

Divide $z$ by $1$ .

$z = \frac{2 x e^{2 x}}{e^{2 x}} + \frac{- e^{2 x}}{e^{2 x}} + \frac{C}{e^{2 x}}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Simplify each term.

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Step 7.3.1.1

Cancel the common factor of $e^{2 x}$ .

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Step 7.3.1.1.1

Cancel the common factor.

$z = \frac{2 x e^{2 x}}{e^{2 x}} + \frac{- e^{2 x}}{e^{2 x}} + \frac{C}{e^{2 x}}$

Step 7.3.1.1.2

Divide $2 x$ by $1$ .

$z = 2 x + \frac{- e^{2 x}}{e^{2 x}} + \frac{C}{e^{2 x}}$

Step 7.3.1.2

Cancel the common factor of $e^{2 x}$ .

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Step 7.3.1.2.1

Cancel the common factor.

$z = 2 x + \frac{- e^{2 x}}{e^{2 x}} + \frac{C}{e^{2 x}}$

Step 7.3.1.2.2

Divide $- 1$ by $1$ .

$z = 2 x - 1 + \frac{C}{e^{2 x}}$